Preconditioning Steady-State Navier--Stokes Equations with Random Data

Abstract

We consider the numerical solution of the steady-state Navier--Stokes equations with uncertain data. Specifically, we treat the case of uncertain viscosity, which results in a flow with an uncertain Reynolds number. After linearization, we apply a stochastic Galerkin finite element method, combining standard inf-sup stable Taylor--Hood approximation on the spatial domain (on highly stretched grids) with orthogonal polynomials in the stochastic parameter. This yields a sequence of nonsymmetric saddle-point problems with Kronecker product structure. The novel contribution of this study lies in the construction of efficient block triangular preconditioners for these discrete systems, for use with GMRES. Crucially, the preconditioners are robust with respect to the discretization and statistical parameters, and we exploit existing deterministic solvers based on the so-called pressure convection-diffusion and least-squares commutator approximations.